How to solve multidimensional input variables in Sobol (M > 20)?

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Hello all,
Recently, I am working on a practical case based on Sobol sensitivity analysis, The response model and probability input model are as follows:
Y=((-X1+X19-8X24-X13+450(X11+X17)+8*(X12+X6+X18)-X7).^2+(-X2+X20+200X24-X14+450(X10+X16)+400*(X12+X6+X18)-X8).^2+(X21+8X22-200X23-X3-X15-8*(X10+X4+X16)-400*(X11+X5+X17)-X9).^2).^(0.5);

InputOpts.Marginals(1).Name = ‘X1’;
InputOpts.Marginals(1).Type = ‘uniform’;
InputOpts.Marginals(1).Parameters = [0 0.029206];
InputOpts.Marginals(2).Name = ‘X2’;
InputOpts.Marginals(2).Type = ‘uniform’;
InputOpts.Marginals(2).Parameters = [0 0.0364];
InputOpts.Marginals(3).Name = ‘X3’;
InputOpts.Marginals(3).Type = ‘uniform’;
InputOpts.Marginals(3).Parameters = [0 0.011321];
InputOpts.Marginals(4).Name = ‘X4’;
InputOpts.Marginals(4).Type = ‘uniform’;
InputOpts.Marginals(4).Parameters = [0 6.0629E-5];
InputOpts.Marginals(5).Name = ‘X5’;
InputOpts.Marginals(5).Type = ‘uniform’;
InputOpts.Marginals(5).Parameters = [0 3.052E-5]; InputOpts.Marginals(6).Name = ‘X6’;
InputOpts.Marginals(6).Type = ‘uniform’;
InputOpts.Marginals(6).Parameters = [0 4.5717E-5];
InputOpts.Marginals(7).Name = ‘X7’;
InputOpts.Marginals(7).Type = ‘uniform’;
InputOpts.Marginals(7).Parameters = [0 0.020339];
InputOpts.Marginals(8).Name = ‘X8’;
InputOpts.Marginals(8).Type = ‘uniform’;
InputOpts.Marginals(8).Parameters = [0 0.048608];
InputOpts.Marginals(9).Name = ‘X9’;
InputOpts.Marginals(9).Type = ‘uniform’;
InputOpts.Marginals(9).Parameters = [0 0.011826];
InputOpts.Marginals(10).Name = ‘X10’;
InputOpts.Marginals(10).Type = ‘uniform’;
InputOpts.Marginals(10).Parameters = [0 6.3456E-5];
InputOpts.Marginals(11).Name = ‘X11’;
InputOpts.Marginals(11).Type = ‘uniform’;
InputOpts.Marginals(11).Parameters = [0 4.31E-5];
InputOpts.Marginals(12).Name = ‘X12’;
InputOpts.Marginals(12).Type = ‘uniform’;
InputOpts.Marginals(12).Parameters = [0 5.7912E-5];
InputOpts.Marginals(13).Name = ‘X13’;
InputOpts.Marginals(13).Type = ‘uniform’;
InputOpts.Marginals(13).Parameters = [0 0.013859];
InputOpts.Marginals(14).Name = ‘X14’;
InputOpts.Marginals(14).Type = ‘uniform’;
InputOpts.Marginals(14).Parameters = [0 0.047617];
InputOpts.Marginals(15).Name = ‘X15’;
InputOpts.Marginals(15).Type = ‘uniform’;
InputOpts.Marginals(15).Parameters = [0 0.010174];
InputOpts.Marginals(16).Name = ‘X16’;
InputOpts.Marginals(16).Type = ‘uniform’;
InputOpts.Marginals(16).Parameters = [0 5.8166E-5];
InputOpts.Marginals(17).Name = ‘X17’;
InputOpts.Marginals(17).Type = ‘uniform’;
InputOpts.Marginals(17).Parameters = [0 3.4953E-5];
InputOpts.Marginals(18).Name = ‘X18’;
InputOpts.Marginals(18).Type = ‘uniform’;
InputOpts.Marginals(18).Parameters = [0 5.6287E-5];
InputOpts.Marginals(19).Name = ‘X19’;
InputOpts.Marginals(19).Type = ‘uniform’;
InputOpts.Marginals(19).Parameters = [0 0.014194];
InputOpts.Marginals(20).Name = ‘X20’;
InputOpts.Marginals(20).Type = ‘uniform’;
InputOpts.Marginals(20).Parameters = [0 0.025789];
InputOpts.Marginals(21).Name = ‘X21’;
InputOpts.Marginals(21).Type = ‘uniform’;
InputOpts.Marginals(21).Parameters = [0 0.048935];
InputOpts.Marginals(22).Name = ‘X22’;
InputOpts.Marginals(22).Type = ‘uniform’;
InputOpts.Marginals(22).Parameters = [0 5.662E-5];
InputOpts.Marginals(23).Name = ‘X23’;
InputOpts.Marginals(23).Type = ‘uniform’;
InputOpts.Marginals(23).Parameters = [0 4.6306E-5];
InputOpts.Marginals(24).Name = ‘X24’;
InputOpts.Marginals(24).Type = ‘uniform’;
InputOpts.Marginals(24).Parameters = [0 4.4639E-5];

The analysis results show some errors:

(a) The estimation of PCE coefficients stopped at polynomial degree 5 and qNorm 1.00 for output variable 1
Final LOO error estimate: 4.281862e-01

(b) Warning: LRA based indices can’t be efficiently computed for large dimensional problems (M>20)
switching back to MCS Sobol indices estimators…

However, I have 24 input variables. How can I solve this problem?
In addition, I don’t quite understand metaopts.expdesign, sampling type: the default option is MC. If I want to change to Sobol sequence sampling, which file should I modify?

I hope to hear from you at your convenience!
Best regards
FANFAN

2

Hi @FANFAN,

I will try to answer your questions regarding the PCE. I am not so practical with the LRA :slight_smile:

The estimation of PCE coefficients stopped at polynomial degree 5 and qNorm 1.00 for output variable 1: you can set the metamodel such that it will not stop at the first degree with a lower LOO error. You can set this as

metaOpts.DegreeEarlyStop = false;

By default, the value is set to true, and it will stop the search at the local minimum of the error. If you need more support, maybe you can paste part of your code.

The experimental design: once you define your input, as you wrote in your question, you need to specify the input object

myInput = uq_createInput(InputOpts);

and then you can collect a sample from it, with the sampling technique you prefer, as

X = uq_getSample(myInput,<method>);
% where <method>: 'mc','lhs','Sobol', or 'Halton'

I hope it was helpful :slight_smile: